Odd-particle systems in the shell model Monte Carlo: circumventing a sign problem

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by  Abhishek Mukherjee

Odd-particle systems in the shell model Monte Carlo: circumventing a sign problem
Abhishek Mukherjee and Y. Alhassid
Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, CT 06520
(Dated: January 17, 2012)
We introduce a novel method within the shell model Monte Carlo approach to calculate the
ground-stateenergyofafinite-sizesystemwithanoddnumberofparticlesbyusingtheasymptotic
behaviorof theimaginary-time single-particle Green’s functions. Themethod circumventsthesign
problemthatoriginatesfromtheprojectiononanoddnumberofparticlesandhashampereddirect
application oftheshell modelMonteCarlo methodtoodd-particlesystems. Weapply thismethod
to calculate pairing gaps of nuclei in the iron region. Our results are in good agreement with
experimental pairing gaps.
2 PACSnumbers: 21.60.Ka,21.60.Cs,21.60De,21.10.Dr,27.40.+z,27.40.+e,26.50.+x
1
0
2 Introduction. The shell model Monte Carlo (SMMC) imaginary-time propagator, e−βH = D[σ]G(σ)Uσ(β),
R
approach[1–4]hasbeenusedsuccessfullytocalculatesta- where β is the inverse temperature, H is the Hamilto-
n
a tisticalpropertiesofnuclei[5–7]withintheframeworkof nian,D[σ]istheintegrationmeasure,G(σ)isaGaussian
J the configuration-interaction shell model. Recently, this weight, and U (β) is the propagator of non-interacting
σ
6 method has also been applied to trapped cold atom sys- nucleons moving in external auxiliary fields σ that de-
1 tems [8, 9]. The SMMC method enables calculations in pendontheimaginarytimeτ (0≤τ ≤β). Thecanonical
model spaces that are many orders of magnitude larger thermalexpectationvalue ofanobservableOˆ isgivenby
]
h thanthosethatcanbetreatedbyconventionaldiagonal- hOˆi = D[σ]G(σ)Tr [OˆU (β)]/ D[σ]G(σ)Tr U (β),
A σ A σ
t ization methods. where TRr denotes a trace over thRe subspace of a fixed
- A
cl For typical effective nuclear interactions, the SMMC number of particles A. In actual calculations we project
u method breaks downat low temperatures because of the on both proton number Z and neutron number N, and
n so-called fermionic sign problem, leading to large statis- in the following A will denote (Z,N).
[ tical errors. In the grand-canonical ensemble the sign Fora quantity Xσ that depends onthe auxiliaryfields
1 problem can be avoidedby constructing good-signinter- σ, we define
v actions that include the dominant collective components
1 of effective nuclear interactions [10]. The remaining part X ≡ R D[σ]|W(σ)|XσΦσ, (1)
4 σ D[σ]|W(σ)|Φ
ofthe effective interactioncanbe accountedfor by using σ
3 R
3 the method of Ref. 2.
where W(σ) = G(σ)Tr U and Φ = W(σ)/|W(σ)| is
A σ σ
. In finite-size systems, such as nuclei, it is necessary
1 the sign. With this definition, the above thermal expec-
0 to use the canonical ensemble, in which the number of tation of an observable Oˆ can be written as hOˆi=hOˆi ,
2 particles is fixed. This particle-number projection gives σ
1 rise to an additional sign problem when the number of where hOˆiσ = TrA[OˆUσ(β)]/TrAUσ(β). In SMMC we
: choose M samples σ according to the weight function
v particles is odd, leading to a rapid growth of statistical k
|W(σ)|, and estimate the average quantity in (1) by
i errors at low temperatures even for good-sign interac-
X X ≈ X Φ / Φ .
tions. Consequently, it has been a major challenge to σ Pk σk σk Pk σk
r For an even number of particles with a good-sign in-
a make accurate estimates for the ground-state energy of
teraction, the averagevalue of the sign Φ remains close
odd-particle systems in SMMC. Accurate ground-state σ
to 1. However, when the number of particles is odd, the
energies are necessary for the calculation of level densi-
average sign decays towards zero as the temperature is
tiesandpairinggaps(i.e.,odd-evenstaggeringofbinding
lowered. Thisleadstorapidlygrowingerrors,hampering
energies).
the direct application of SMMC at low temperatures for
Here we develop a method based on the asymptotic
odd-particle systems.
behavior of the imaginary-time single-particle Green’s
For a rotationally invariant and time-independent
functions ofaneven-particlesystemto calculateground-
Hamiltonian, we define the following scalar imaginary-
state energies of neighboring odd-particle systems. This
time Green’s functions [12]
methodis somewhatsimilarin spiritto a techniqueused
in lattice quantum chromodynamics to extract hadron Tr e−βHT a (τ)a† (0)
masses(see,e.g.,inRef.11). WeapplyourGreen’sfunc- Gν(τ)= A(cid:2) TrPme−βνHm νm (cid:3), (2)
tionmethodtocalculatepairinggapsofnucleiintheiron A
region using the complete fp+g9/2 shell model space. where ν ≡(nlj)labels the nucleonsingle-particleorbital
Green’s functions in SMMC. The SMMC method is with radial quantum number n, orbital angular momen-
basedontheHubbard-Stratonovichrepresentationofthe tum l and total spin j. Here T denotes time ordering
2
and a (τ) ≡ eτHa e−τH is an annihilation operator Assuming A is an even-evennucleus, A ≡(Z,N±1)
νm νm ±
of a nucleon at imaginary time τ (−β ≤ τ ≤ β) in a areneighboringodd-evennucleiwithoddnumberofneu-
single-particle state with orbital ν and magnetic quan- trons. WedenotebyJn(n=0,1,2,...)then-thexcited
tum number m (−j ≤m≤j). state with total spin J and define the energy differences
Using the Hubbard-Stratonovich transformation, the ∆E (A ) = E (A )−E (A), where E (A) is the
J ± J0 ± 00 Jn
Green’s functions defined in(2) canbe writtenin a form energy of the state Jn with particle number A. Assum-
suitable for SMMC calculations ing that the ground state of the even-even nucleus has
spinzero,∆E (A )is the energydifferencebetweenthe
J ±
[U (τ)(I−hρˆi ] for τ >0 lowest state of a given spin J in the odd-even nucleus
X σ σ νm,νm
Gν(τ)= m , AAs±suamnidngthtehagtrtohuendgrostuantde sotfatteheofetvheen-oedvde-nevneunclneuucsleAus.
hρˆi U−1(|τ|) for τ ≤0 A is J = j, where j is one of the single-particle or-
Xm (cid:2) σ σ (cid:3)νm,νm (3) bEi±ta(lAsp)in=vaElue(sA, )it+s c∆orEresp(oAndi)n,gwehneerregy∆Eis giveisntbhye
gs ± 00 min ± min
where we have used the notation in Eq. (1). Here Uσ(τ) minimum of ∆Ej(A±) over the possible values of j.
and I are matrices in the single-particle space repre-
senting the propagator U (τ) and the identity, respec-
σ
tively. hρˆiσ isa matrixinthe single-particlespacewhose The neutron Green’s function Gν(τ) that corresponds
νm,ν′m′ matrix element hρˆνm,ν′m′iσ is defined in terms to an orbital with angular momentum j can be written
of the one-body density operator ρˆνm,ν′m′ =a†ν′m′aνm. as
 
Gν(τ)=C(β)e−∆Ej(A±)|τ|1+ X RJJn′n′(A±,ν)e−|τ|[EJ′n′(A±)−Ej0(A±)]e−(β−|τ|)[EJn(A)−E00(A)] (4)
 Jn6=00 
 J′n′6=j0 
where the + (−) subscript should be used for τ > a few tens of keV, which is comparable to our target ac-
0 (τ ≤ 0) and C(β) is a τ-independent con- curacy. Forlow-andmedium-mass nuclei,we expectthe
stant. RJ′n′(A ,ν) are scaled weights defined by energydifferences to be &1 MeV andthe scaledweights
Jn ±
RJ′n′(A ,ν) = |(J′n′||a†||Jn)|2/|(j0||a†||00)|2 and to be much smaller than one. Thus, calculations with β
Jn + ν ν
RJ′n′(A ;ν) = |(J′n′||a ||Jn)|2/|(j0||a ||00)|2, where of a few MeV−1 and with an asymptotic regime of τ ∼1
Jn − ν ν
(J′n′||a†||Jn) and (J′n′||a ||Jn) are reducedmatrix ele- MeV should be sufficient. This can be validated explic-
ν ν
ments of a† and a between the state Jn in A and the itlyinsd-shellnuclei(seebelow),whoseHamiltoniancan
ν ν
states J′n′ in A and A , respectively. be diagonalized numerically. For larger model spaces, it
+ −
is not possible to calculate explicitly the corrections in
When all terms in the summation on the r.h.s. of Eq.
the sum of Eq. (4), and the asymptotic region has to be
(4) are small, the Green’s function can be well approxi-
determinedbythegoodnessofthelinearfitstolnG (τ).
matedbyasingleexponential,Gν(τ)∼e−∆Ej(A±)|τ|. In ν
thisasymptoticregimeforτ,wecancalculate∆E (A ), Results. We first tested the Green’s function method
j ±
and hence E (A ) from the slope of lnG (τ). This is in sd-shell nuclei and then applied it to medium-mass
gs ± ν
themethodweuseheretocalculatethe ground-stateen- nuclei in the complete (pf +g9/2) shell. In these nuclei,
ergy of odd-A nuclei with odd number of neutrons. The we carried out calculations for several values of β in the
ground-state energy of odd-A nuclei with odd number range 3 MeV−1 ≤ β ≤ 4 MeV−1. For each β, we calcu-
of protons can be similarly calculated using the proton lated Gν(τ) for a range of values of τ in steps of 1/32
Green’s functions. MeV−1. We chose the asymptotic region in τ such the
linear fits to lnG (τ) have a χ2 per degree of freedom
In principle, the asymptotic regime is accessed in the ν
∼ 1 or less in all cases considered. We find that a good
limit β → ∞. However, in a shell-model Hamiltonian
asymptotic region is 0.5 MeV−1 ≤τ ≤2 MeV−1.
with discrete, well separated energy levels, only a few
transitions give significant contributions. If the relative Within the asymptotic region, we fit a straight line to
contribution from the sum in Eq. (4) is less than a few lnG for each possible subset of points in τ for which
ν
percent, then (assuming that |τ| ∼ 1 MeV) the sensitiv- G (τ) has been calculated. The mean and standard de-
ν
ity of the slope of lnG (τ) to this contribution is about viation of the slopes so obtained are used to estimate
ν
3
-200
30
0.2 58Fe
0 -202
|1p3/220 -0.2 |τ| 1(MeV-1) 2 eV)
G M 1
|ln 0.1 56Fe E (-204 MeV)
10 (E0.1
0 σ
0.01
-0.1 -206 3.2 3.6 4
|τ| 1(MeV-1) 2 β (MeV-1)
0
0 1 2 3 3 3.2 3.4 3.6 3.8 4
|τ| (MeV-1) β (MeV-1)
FIG. 1: The absolute valueof logarithm of theGreen’s func- FIG. 2: The energy of the 57Fe nucleus calculated from the
tion(2)fortheneutronorbitalν =1p3/2in56Fe(lowercurve, present method and direct SMMC are shown by solid and
τ >0) and 58Fe (upper curve, τ ≤0) at β =4 MeV−1. The opensquares,respectively. Theerrorbarsdescribethestatis-
solidbluelinesarelinearfitsfor0.5MeV−1 ≤|τ|≤2MeV−1. ticalerrors. Inset: thestatisticalerrorsfortheenergyof57Fe
The insets show the deviations from these linear fits. in the present method (solid squares) and in direct SMMC
calculations (opensquares) areshownon alogarithmic scale.
The statistical errors for the energy of 56Fe using the same
∆E (A )anditsstatisticalerror,respectively,ateach Hamiltonian are shown by open circles.
min ±
β. A weighted average of the results at different values
of β is then taken.
In a few selected cases,we alsoperformed calculations particle systems suffers from a sign problem which leads
forlargervaluesofβ (i.e., β >4MeV−1),andfoundthe to very large statistical errors at low temperatures. In
correspondingvaluesof∆E (A )tobeconsistentwith contrast, the method presented here does not have such
min ±
those obtained in the region 3 MeV−1 ≤ β ≤ 4 MeV−1. problem. This is illustrated in Fig. 2 where we com-
This indicates that for the model spaces and particle pare the energy and its statistical error for the 57Fe nu-
numbers considered, the above chosen values of β are cleus in the present method (using the neutron Green’s
sufficiently large to isolate the ground state of the corre- functions of 56Fe) with the results obtained from the di-
sponding even-even nucleus. rect method. The errors in the present method remain
roughlyconstantwithβ. Atβ =3MeV−1 thestatistical
Foragivenoddsystem(anodd-evennucleus)thereare
error in the direct method is about 5 times larger than
twoneighboringevensystems(even-evennuclei),andour
the presentmethod while at β =4 MeV−1 it is about20
method can be used by starting from either of the even
times larger. The inset shows the statistical errors on a
systems. Unless noted otherwise, the results we report
logarithmic scale. For comparison we have also included
here are the averageof both of these calculations.
thestatisticalerrorintheenergyoftheeven-evennucleus
To test the validity and accuracy of our method, we
56Fe using the same Hamiltonian.
performed calculations in the sd shell using a schematic
good-sign Hamiltonian. In all cases, our results devi- We applied our Green’s function method for fami-
ated no more than 0.1%from the exactground-state en- lies of odd-neutron isotopes: 47−49Ti, 51−57Cr, 53−61Fe,
ergies, obtained by diagonalizing the Hamiltonian with 59−65Ni, 63−67Zn and 71−73Ge. The ground-state spins
the OXBASH code [13]. For example for 29Si we found a we determine are in agreementwith experimental values
ground-state energy of −133.98± 0.04 MeV compared inallcasesexceptfor47Ti,57Feand63Ni. Theanomalous
with the exact result of −133.95 MeV. Our method also ground-state spin of 57Fe from the shell model perspec-
reproduced correctly the ground-state spin in all cases. tive is well documented in the literature [14].
Weappliedourmethodtonucleiinthe(pf+g )shell, Inourmethodweextractdirectlytheodd-evenground
9/2
using the isospin-conserving Hamiltonian of Ref. [5]. stateenergydifferences,andthereforethismethodispar-
Typical results are demonstrated in Fig. 1, in which the ticularlysuitableforaccuratecalculationsofpairinggaps
absolute value of the logarithm of the Green’s functions (i.e., odd-even staggering of masses).
for the neutron orbital ν = 1p3/2 in 56Fe (τ > 0) and Whenextractinganodd-evenground-stateenergydif-
in 58Fe (τ ≤ 0) are plotted versus |τ| for β = 4 MeV−1. ferencesuchas∆E (A )weusetheHamiltonianofthe
min +
The linear fits (solid lines) were used in the calculation A nucleusforboththeA andAnuclei. Sincethefp+
+ +
of the ground-state energy of 57Fe. The deviations from g -shellHamiltonianweuseisnucleus-dependent[5],it
9/2
the linear fits are shown in the insets of Fig. 1. is necessary to correct the ground-state energy of the A
Adirectapplicationofthe SMMC methodtothe odd- nucleus. As the latter is an even-even nucleus, this cor-
4
48 50 52 54 54 56 58 60 circles), where they are compared with the experimental
pairing gaps (open circles) as determined from odd-even
2
staggering of binding energies. Our results agree quite
1.5 well with the experimental values; in most cases the de-
viation of the theoretical pairing gaps from their experi-
1 Ti Cr Fe mental counterparts is less than 15%. Systematic devia-
tionsareobservedfortheironisotopesaboveA=59and
)0.5
V forthegermaniumisotopes. Forthegermaniumisotopes
Me the size of the model space might be insufficient, while
( thedeviationfortheironisotopesindicatesthenecessity
n
∆ 2 to refine our isospin-conserving Hamiltonian.
Conclusion. We have described a practical method
1.5
that circumvents a sign problem for calculating the
1 Ni Zn Ge ground-state energy of odd-particle systems in the shell
Exp. model Monte Carlo approach. We have demonstrated
0.5
Theory the usefulness of the method by calculating pairing gaps
of nuclei in thefp + g shell. This method can also
9/2
60 62 64 A 64 66 68 70 be applied to other finite-size many body systems such
as trapped cold atoms. In principle this method can be
FIG.3: Neutronpairinggaps∆ asafunctionofmass num- used more generally to calculate the lowest energy state
n
berA in fp+g9/2-shell nuclei. The gaps calculated with the for a given spin. However, when such a state is an ex-
present Green’s function method (solid circles connected by citedstate,the statisticalerrorsarelargeranditis more
solid lines) are compared with the experimental gaps (open
difficult to identify the asymptotic regime.
circles connectedbydashedlines). Thetheoreticalstatistical
errors are smaller than thesize of the symbols. Acknowledgements. This work was supported in part
bythe U.S.DepartmentofEnergyGrantNo. DE-FG02-
91ER40608. Computational cycles were provided by the
rection can be found in direct SMMC calculations for
HighPerformanceComputing CenteratYale University.
the A nucleus. However,this correctioncan also be esti-
mated as follows. The dependence of the interaction on
thenucleusisratherweak;thestrengthsofthemultipole-
multipole interactions depend weakly on the mass num-
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W. E. Ormand, Phys.Rev. Lett.72, 613 (1994).
ε (A)oftheorbitalsµ. Correctingforthisvariation,the
µ [3] S. E. Koonin, D. J. Dean, and K. Langanke, Phys. Rep.
neutronseparationenergyfortheA+ nucleusisgivenby 278, 1 (1997).
[4] Y. Alhassid, Int.J. Mod. Phys. B 15, 1447 (2001).
Sn(A+)=−∆Emin(A+)+X[εµ(A)−εµ(A+)]hnµiA , [5] H. Nakada and Y. Alhassid, Phys. Rev. Lett. 79, 2939
µ (1997).
(5) [6] Y.Alhassid,S.Liu,andH.Nakada,Phys.Rev.Lett.83,
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+
second term on the r.h.s. of (5) approximates the differ- 101, 082501 (2008).
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K. Langanke, Phys. Rev.A 80, 013613 (2009).
when calculated using the respective Hamiltonians for
[9] C. N.Gilbreth and Y.Alhassid, in preparation.
theAandA nuclei. Weverified(insd-shellnuclei)that
+ [10] M. Dufour and A. P. Zuker, Phys. Rev. C 54, 1641
this approximation is highly accurate and well within a
(1996).
typical statistical error. In our calculations we used (5) [11] R.Gupta,inProbingthestandardmodelofparticleinter-
since the resulting statistical error is much smaller than actions, LXVIII Les Houches Summer School, edited by
the statistical error of direct SMMC calculations. R. Gupta, A. Morel, E. de Rafael and F. David (North-
The neutron separation energy for the A nucleus is Holland, Amsterdam, 1999); arXiv:hep-lat/9807028.
[12] Ingeneral,wecanconstructsingle-particleGreen’sfunc-
given by a similar expression. The neutron pairing gaps
tions that are tensors of rank K (0≤K ≤2j) but only
canthen be calculatedfromthe differences ofseparation
the scalar K =0 Green’s function is non-vanishing.
energies ∆ (A) = (−)N[S (A ) − S (A)]/2, where A
n n + n [13] B. A. Brown, A. Etchegoyen, and W. D. M. Rae, MSU-
can now be either an even-even or an odd-even nucleus. NSCL Report No. 524 (1988).
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5